Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods.
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22 views
Constrained computational optimization of a functional of a vector valued function.
I am trying to increase the efficiency of a program I have written that must run in real time. I am asking this question in a broad sense, since I'm not sure what tools are available to me.
I am ...
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0answers
10 views
Tools for optimizing asymptotic bounds.
Is there any tool for this task ?
Given the asymptotic bound in term of $n$ and other paramaters $t_1,\dots,t_r$, then return the value for each $t_i$ which optimizes the expression in term of $n$, ...
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1answer
86 views
Derivative of Associated Legendre polynomials at $x = \pm 1$
I'm creating meshes for spherical harmonics, and I need a normal at a given point. Whenever I'm at the poles, $\cos{\theta} = \pm 1$, and I do not know how to find the derivative there. All the ...
4
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1answer
43 views
About parallel time computation
I am studying a paper where it is mentioned that Newton iteration may be used to compute the inverse of $n \times n$, well- conditioned matrix in parallel time $o(\log^2n)$ and that this computation ...
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2answers
56 views
What free software can I use to solve a system of linear equations containing an unknown?
Question: What free software can I use to solve a system of linear equations $M\mathbf{x}=\mathbf{y}$ where the entries of $\mathbf{y}$ vary with an unknown quantity $n$?
Presumably I could do ...
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0answers
45 views
Simpson's rule characteristics
I just wanted to ask a quick question in regards to simpson's rule for integration. I have been reading up on the trapezoidal rule, and have found the notations and have an understanding such that:
...
7
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2answers
98 views
Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$
I realize there is probably not a closed form, but is there an efficient way to calculate the following expression?
$$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$
I've noticed $$\sum_{k=1}^n ...
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0answers
100 views
C++ Polynomial Multiplication [closed]
\begin{eqnarray} \text{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~} \end{eqnarray}\begin{eqnarray}
\text{IF YOU HAVE A QUESTION, THEN ASK AND I CAN ...
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46 views
How to find an expression whose value is 190
Given a set of numbers (in this case):
3, 7, 7, 100, 50
Either:
prove it is impossible to form the number k = 190 using ( ) + - * / operators between sub set of the these numbers ex: 1000 = ((3 + ...
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1answer
48 views
how can I find the following problem using laplace transform?
For example here is the problem: $(t^2 \cos{\omega t})u(t)$
I have to find it using laplace transform; here is what I think it is,
I have $t^2$$(\cos{\omega t})u(t)$ which I think I can solve them ...
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5answers
525 views
What is the value of $2^{3000}$
What is the value of $2^{3000}$?
How to calculate it using a programming language like C#?
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0answers
49 views
Evaluation of a slow continued fraction
Puzzle question... I know how to solve it, and will post my solution if needed; but those who wish may participate in the spirit of coming up with elegant solutions rather than trying to teach me how ...
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1answer
46 views
How are 10-20 digit multiperfect and hemiperfect numbers efficiently computed?
This numericana item on multiperfect and hemiperfect numbers contains some impressively enormous numbers. How were these actually computed ? The associated OEIS pages (A007691 & A159907) just ...
3
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1answer
62 views
What is the fastest computational graph theory package?
What is the fastest computational graph theory package with respect to executing algorithms and computing graph theoretic data?
I am aware of this related question, which requests graph theory ...
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1answer
48 views
How do I determine if two of my software's representation of algebraic numbers are equal?
I have software which stores information about algebraic numbers with absolute precision. If you build it up by creating instances of a Python representation of an integer, float, Decimal, or string, ...
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2answers
38 views
Does rationalizing the denominator lead to more or less round-off error?
I evaluated $\frac{1}{\sqrt{2}}$ and $\frac{\sqrt{2}}{2}$ in Matlab, and got a slight difference: $0.707106781186547$ and $0.707106781186548$, respectively. Which is more accurate, the one with the ...
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1answer
43 views
Matrix completion: supplementary questions
Continuation of the question here, what is going to happen if we change the some of the conditions. I write it as a quote from here and change the appropriate places which are underlined:
I need ...
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3answers
86 views
Drawing graphs (vertices and edges) with or without technology
Given a collection of vertices $V$ and a collection of edges $E \subseteq V\times V$, is there an algorithm or program that will allow you to draw a nice graph? The placing of the vertices is very ...
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1answer
78 views
Best graphing program for Mac or PC?
I just bought the highest end iMac, with a student discount, of course, and was wondering what is the best graphing program out there. A program that can graph any equation that I throw at it AND one ...
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2answers
88 views
How to get the minimum angle between two crossing lines?
I'm not a student, I'm just a programmer trying to solve a problem ... I just need the practical way to calculate the smallest angle between two lines that intersect. The value, of course, must always ...
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1answer
91 views
Matrix completion
I need to find an algorithm (if exists) of the following matrix completion problem.
I need to construct $n^2$ positive semi-definite matrices, say $\{P_i\}_{i=1}^n$. Entries of these matrices are ...
1
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1answer
80 views
Minimizing mean squared error
I want to find a $d$ that minimizes the value of the expression below. I think the first step is to find the derivative w.r.t. $d$ (is that correct? If not, what is the first step?). If so, I'm having ...
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3answers
117 views
Mathematical Limitations of Computer Experiments
One problem that has always bothered me is the limitations of computers in studying math. With a chaotic dynamical system, for example, we know mathematically that they possess trajectories that never ...
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0answers
61 views
Drawing finite state machines based on regular expressions
I have problems drawing the following finite state machines which use the {0,1} alphabet:
Are these DFAs or NFAs? How to determine the latter?
Tyvm for your time and help!
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0answers
49 views
Solving an overdetermined system of inequalities using null-space arguments
The solutions to a linear system of equations:
$$A\cdot x = b$$
(where $x$ is a $(n\times 1)$ column vector, $b$ is a $(m\times 1)$ column vector and $A$ is $(m\times n)$ matrix)
can all be ...
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1answer
36 views
Discrete numerical derivative with respect to d/d(n*x)
How can I generate a stencil for a d/d(n*x) operator?
I am writing a program that needs a method to calculate line derivatives in an image. If we want to calculate the simplest forward derivative ...
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0answers
35 views
Qubit state finding
Suppose we have two qubits in the state $x|00\rangle+y|11\rangle $.
What is the resulting state of the second qubit in that case? Use and to denote and respectively.
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2answers
58 views
Can product of all pairwise sums be computed faster than the naive method?
Let $S$ be a set of integers. $|S|=n$.
Can we find the product $\prod_{a,b\in S} a+b$ faster than naively add all pairs then multiply them one by one? By faster, I mean use less than $O(n^2)$ ...
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1answer
49 views
How to find a close form expression in terms generating functions for the triple summation
Given $$\sum\limits_{i=0}^\infty a_i z^i=A(z)$$ and $$\sum\limits_{i=1}^\infty b_i z^i=B(z)$$ and $$\sum\limits_{i=0}^\infty c_i z^i=C(z)$$ Find $\sum\limits_{i=1}^\infty\sum\limits_{j=0}^\infty a_j ...
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votes
6answers
557 views
Fastest Square Root Algorithm
What is the fastest algorithm for finding the square root of a number?
I created one that can find the square root of "987654321" to 16 decimal places in just 20 iterations (I'm not ready to release ...
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1answer
33 views
Amicable numbers
Def: a pair natural numbers $a$, $b$, $a\ne b$ are an Amicable pair if
$\sum_{d|a,a\ne d}d = b$ and $\sum_{d|b, b\ne d}d = a$.
Ok. So I'm trying to optimize a calculation for finding the number of ...
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3answers
124 views
Understanding recursive definitions of a language.
I am having difficulty understanding the recursive definition of a language. The problem asked how to write this non recursively. But I want to understand just how a recursive definition of a ...
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0answers
36 views
Function design and Super-Symmetry Test Requirements
I request everyone concerned to give a serious thought and respond.
If you are asked to design two independent functions say S() and P() and should satisfy the following conditions:
There will be ...
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1answer
39 views
Probability : Dividing a list into 2 classes
I have a list of integer numbers ($n$). I am dividing it into two parts $n_1$ (smaller) and $n_2$ (bigger) such that the length of $n_1 \ge a*n$; $a$ is positive and $a \lt 0.5$. What is the ...
4
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1answer
171 views
A search for integers which can be written as a sum of two squares in multiple ways
As part of a number theory hobby project, I'm looking for a computational way to enumerate all integers $n$ which can be written as a sum of two integer squares in three or more ways. The range of ...
6
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1answer
100 views
Fractional part of exp(x)
I have a real number $x$ (for concreteness, say $10^4<x<10^6$) and would like to find $e^x-\lfloor e^x\rfloor$ to reasonable precision (10-20 decimal places). What is the most efficient method?
...
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0answers
77 views
How to convert a hologram into an image?
Suppose one knows in full detail the phase and intensity of monochromatic light in a plane. This is basically what a hologram records, at least for some section of a plane. By using this as the ...
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202 views
Simple Lanczos algorithm code to obtain eigenvalues and eigenvectors of a symmetric matrix
I would like to write a simple program (in C) using Lanczos algorithm. I came across a Matlab example which helped me to understand a bit further the algorithm, however from this piece of code I can't ...
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0answers
70 views
How to make normalized cross correlation robust to small changes in uniform regions
the problem is described below:
Given 2 sets of data: A= { 91 87 85 85 84 90 85 83 86 86 90 86 84 89 93 87 89 91 95 97 91 92 97 101 101 },
B = {133 130 129 131 133 136 131 131 135 135 133 133 133 ...
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2answers
92 views
Number of ways to move 1 or more elements from one list to the previous list until one list remains
Given N elements, divided into at most N groups, which are then labeled 1 thru N, move all of the elements into the group labeled 1. By moving 1 to all of the elements, in group i to i-1. This means ...
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2answers
230 views
Questions that can be solved using Excel.
I recently started to realize that Excel is a powerful tool that can solve many problems. What interesting mathematics problems are there can be solved using excel? I am looking for a set of ...
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1answer
50 views
Where does the input x in Turing Machine subroutines come from in solving reductions to undecidable problems?
I'm taking an introduction to computation theory class and we went over the chapter on undecidable problems and proving undecidability through reductions. I can't seem to grasp some of the simplest ...
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0answers
60 views
graphing a differential equation using RK4 (making a slider graph in Excel)
So I'm graphing a system of nonlinear differential equations for my Math class and I've begun to calculate the values in Excel using Runge Kutta 4. The system includes a variable "b" that I can change ...
2
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1answer
77 views
Oracle turing machine
I am learning computational complexity and this is a question of my assignment that I have issues trying to solve/understand.
An oracle Turing Machine M with oracle A is a Turing Machine with an ...
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1answer
70 views
Complement of NP-Complete
If a language L is NP-complete, with respect to polynomial time reducibility, does L ≤ co-L in polynomial time?
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16 views
Show that for every deterministic algorithm ALG, there is some scenario in which the total distance John’s trucks traverse under ALG is 3OPT
There are 3 popular beach resorts, A, B, and C, which reside on a line:
A-----(1km)-----B-----(1km)------C.
The distances between the ...
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0answers
20 views
Model to reward adaptative communication
I'm currently working in different simulations focused on the study of artificial evolution of communications in robots, and I have stumbled against the problem of its mathematic formulation.
The ...
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2answers
75 views
floating point binary arithmetic
Prove that the decimal number $\displaystyle \frac{1}{5}$ cannot be represented by a finite expansion in the binary system.
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2answers
76 views
Finite representation in the binary $\implies$ finite representation in the decimal system
Any number that has a finite representation in the binary system have a finite representation in the decimal system. Why?
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1answer
71 views
Simplified way of finding a complex number raised to another complex number
This question here has the answer but I'm still in school and I don't understand any of it. I'm writing a computer program that takes a complex number a + ib and raises it to c + id and I need to ...
